A geometric approach to the Cohen-Lenstra heuristics

We give a new geometric description of when an element of the class group of a quadratic field, thought of as a quadratic form q, is n-torsion. We show that q corresponds to an n-torsion element if and only if there exists a degree n polynomial whose resultant with q is ±1. This is motivated by a more precise geometric parameterization which directly connects torsion in class groups of quadratic fields to Selmer groups of singular genus 1 curves.